Question

the points P ( 4,3) Q ( -5,-1) and T ( 2,-2) are vertices of a
(a) Equilateral triangle
(b) Isosceles triangle
(c) Scalene triangle
(d) No triangle ​

Answer 1

$\boxed{\rm{\orange{Given \longrightarrow }}}$

⇢Points are P(4,3), Q(-5-1), T(2,-2)

$\boxed{\rm{\red{To\:Find\longrightarrow }}}$

⇢whether the given points form a triangle

$\boxed{\rm{\pink{solution \longrightarrow }}}$

⇢First we find the lengths $PQ,QT,PT$

$PQ=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$

$PQ=\sqrt{(4+5)^2+(3+1)^2}$

$PQ=\sqrt{9^2+4^2}$

$PQ=\sqrt{81+16}$

$\bf\,PQ=\sqrt{97}$

$QT=\sqrt{(-5-2)^2+(-1+2)^2}$

$QT=\sqrt{(-7)^2+1^2}$

$QT=\sqrt{49+1}$

$\bf\,QT=\sqrt{50}$

$PT=\sqrt{(4-2)^2+(3+2)^2}$

$PT=\sqrt{2^2+5^2}$

$PT=\sqrt{4+25}$

$\bf\,PT=\sqrt{29}$

$\implies\bf\,PQ{\neq}QT{\neq}PT$

∴The given points P,Q and T form a scalene triangle

$\boxed{\textbf{Option (c) is corrct}}$

Answer 2

$\huge\star\:\:{\orange{\underline{\pink{\mathbf{Solution:-}}}}}$

• Finding the distance between the points,

(PQ)² = (-5-4)² + (-1-3)²

         = (-9)² + (-4)²

         = 81 + 16

         = 97

PQ =$\sqrt{97}$

Now,

(QT)² = (2+5)² + (-2+1)²

         = (7)² + (-1)²

         = 49 + 1

         = 50

QT =$\sqrt{50}$

And,

(PT)² = (2-4)² + (-2-3)²

         = (-2)² + (-5)²

         = 4 + 25

         = 29

PT =$\sqrt{29}$

So,

We can conclude that all side are different

PQ ≠ QT ≠ PT

Hence,

It is a Scalene triangle

Option c) Scalene triangle is correct